التحويلات الهندسية مع دوال القيمة المطلقة - كتاب الرياضيات - الصف 12 - الفصل 1 - المملكة العربية السعودية

تستعمل تحويلات هندسية أخرى غير قياسية تتضمن القيمة المطلقة. --- SECTION: مفهوم أساسي --- مفهوم أساسي التحويلات الهندسية مع دوال القيمة المطلقة --- SECTION: g(x) = |f(x)| --- يغير هذا التحويل الهندسي أي جزء من منحنى الدالة يقع تحت المحور x ليصبح فوقه بالانعكاس حول المحور x. --- SECTION: g(x) = f(|x|) --- يغير هذا التحويل الهندسي أي جزء من منحنى الدالة الموجود إلى يسار المحور y ويضع مكانه صورة جزء المنحنى الواقع إلى يمين المحور y بالانعكاس حول المحور y. --- SECTION: إرشاد تقني --- تحويلات القيمة المطلقة يمكنك التحقق من أثر التحويل الهندسي على منحنى القيمة المطلقة باستعمال الحاسبة البيانية. ويمكنك أيضاً تمثيل كلا الدالتين في المستوى الإحداثي نفسه. --- SECTION: مثال 7 --- مثال 7 --- SECTION: وصف التحويلات الهندسية وتمثيلها --- وصف التحويلات الهندسية وتمثيلها استعمل منحنى الدالة 4x - x³ = (f(x المبين في الشكل 1.5.6 لتمثيل كل من الدالتين الآتيتين بيانياً: --- SECTION: g(x) = |f(x)| (a --- يقع الجزء السالب من منحنى (x)f في الفترتين (2-, ∞-) و (0, 2)؛ لذا يتم عكس هذين الجزأين حول المحور x ويترك الجزء الباقي من المنحنى دون تغيير. --- SECTION: h(x) = f(|x|) (b --- ضع مكان جزء المنحنى الموجود إلى يسار المحور y انعكاس الجزء الموجود إلى يمينه حول المحور y. --- SECTION: تحقق من فهمك --- تحقق من فهمك استعمل منحنى الدالة (x)f في كل من الشكلين أدناه؛ لتمثيل كل من الدالتين |(x)f| = g و |x|)f = h بيانياً: 54 الفصل 1 تحليل الدوال وزارة التعليم Ministry of Education 2025 - 1447 --- VISUAL CONTEXT --- **DIAGRAM**: g(x) = |f(x)| Description: Two graphs illustrating the transformation g(x)=|f(x)|. The top graph shows an original parabola y=f(x) opening downwards, intersecting the x-axis at two points. The bottom graph shows the transformed function g(x)=|f(x)|, where the portion of the original graph below the x-axis has been reflected upwards, making all y-values non-negative. The x-axis and y-axis are labeled, with the origin O marked. X-axis: x Y-axis: y Data: The top graph shows y=f(x) as a parabola opening downwards. The bottom graph shows g(x)=|f(x)| where the parts of the parabola below the x-axis are reflected above the x-axis. Context: Illustrates the effect of taking the absolute value of the entire function, reflecting any part below the x-axis above it. **DIAGRAM**: g(x) = f(|x|) Description: Two graphs illustrating the transformation g(x)=f(|x|). The top graph shows an original parabola y=f(x) opening downwards, intersecting the x-axis at two points. The bottom graph shows the transformed function g(x)=f(|x|), where the part of the original graph to the left of the y-axis has been removed, and the part to the right of the y-axis has been reflected across the y-axis to the left side. The x-axis and y-axis are labeled, with the origin O marked. X-axis: x Y-axis: y Data: The top graph shows y=f(x) as a parabola opening downwards. The bottom graph shows g(x)=f(|x|) where the portion of the parabola to the right of the y-axis is kept, and then reflected across the y-axis to replace the portion that was originally on the left. Context: Illustrates the effect of taking the absolute value of the input variable, resulting in a graph that is symmetric with respect to the y-axis, using the right-hand side of the original function. **GRAPH**: الشكل 1.5.6 Description: A graph of the cubic function f(x) = x³ - 4x. The graph passes through the origin (0,0), has local maximum and minimum points, and intersects the x-axis at approximately x=-2, x=0, and x=2. The x-axis and y-axis are labeled, with the origin O marked. The grid lines are visible. X-axis: x Y-axis: y Data: The graph shows a cubic function with roots at x=-2, x=0, and x=2. It has a local maximum in the second quadrant and a local minimum in the fourth quadrant. Key Values: x-intercepts: -2, 0, 2, y-intercept: 0 Context: This is the original function f(x) = x³ - 4x used as a base for transformations in Example 7. **GRAPH**: g(x) = |f(x)| Description: A graph of the transformed function g(x) = |x³ - 4x|. This graph is derived from f(x) = x³ - 4x (Figure 1.5.6). The portions of the original cubic function that were below the x-axis (in the intervals (-∞, -2) and (0, 2)) have been reflected upwards across the x-axis. All y-values are non-negative. The x-axis and y-axis are labeled, with the origin O marked. The grid lines are visible. X-axis: x Y-axis: y Data: The graph shows the absolute value of the cubic function f(x)=x³-4x. All parts of the original function that were below the x-axis are now reflected above it, resulting in a graph where all y-values are positive or zero. Key Values: x-intercepts: -2, 0, 2, y-intercept: 0 Context: Demonstrates the transformation g(x)=|f(x)| on a cubic function, where negative y-values are made positive by reflection over the x-axis. **GRAPH**: h(x) = f(|x|) Description: A graph of the transformed function h(x) = |x|³ - 4|x|. This graph is derived from f(x) = x³ - 4x (Figure 1.5.6). The portion of the original cubic function to the left of the y-axis has been removed, and the portion to the right of the y-axis has been reflected across the y-axis to create a symmetric graph. The x-axis and y-axis are labeled, with the origin O marked. The grid lines are visible. X-axis: x Y-axis: y Data: The graph shows the transformation h(x)=f(|x|) applied to the cubic function f(x)=x³-4x. The graph is symmetric about the y-axis, with the right half of the original function reflected to the left side. Key Values: x-intercepts: -2, 0, 2, y-intercept: 0 Context: Demonstrates the transformation h(x)=f(|x|) on a cubic function, resulting in a graph symmetric about the y-axis, using the right-hand side of the original function. **GRAPH**: 7A Description: A graph of the rational function f(x) = 5 / (3x - 4). The graph shows two branches, indicating a vertical asymptote at x = 4/3 (approximately 1.33) and a horizontal asymptote at y = 0. The x-axis and y-axis are labeled, with the origin O marked. The grid lines are visible. The function is decreasing on both sides of the vertical asymptote. X-axis: x Y-axis: y Data: The graph shows a rational function with a vertical asymptote at x=4/3 and a horizontal asymptote at y=0. The function values are positive for x > 4/3 and negative for x < 4/3. Key Values: vertical asymptote: x = 4/3, horizontal asymptote: y = 0 Context: This graph is provided for the student to apply the absolute value transformations g(x)=|f(x)| and h(x)=f(|x|) as part of the 'Check Your Understanding' exercise. **GRAPH**: 7B Description: A graph of the absolute value function f(x) = |2 - x|. This is a V-shaped graph, opening upwards, with its vertex at (2, 0). The graph intersects the y-axis at (0, 2). The x-axis and y-axis are labeled, with the origin O marked. The grid lines are visible. The function is decreasing for x < 2 and increasing for x > 2. X-axis: x Y-axis: y Data: The graph shows a V-shaped absolute value function with its vertex at (2,0) and y-intercept at (0,2). Key Values: vertex: (2, 0), y-intercept: (0, 2) Context: This graph is provided for the student to apply the absolute value transformations g(x)=|f(x)| and h(x)=f(|x|) as part of the 'Check Your Understanding' exercise. Note that f(x)=|2-x| is already an absolute value function, so g(x)=|f(x)| would be identical to f(x).